Better examples through peer instruction

I just gave midterm evaluations in my classes, and for the item about “What could we be doing differently to make the class better?”, many students put down: Do more examples at the board. I think I’ve seen that request more often than any other in my classes at midterm. This is a legitimate request (it’s not like they’re asking for free points or an extra day in the weekend), but honestly, I’m hesitant to give in to it. Why? Two reasons.

First, doing more examples at the board means more lecturing, therefore less active learning, and therefore more passivity and dependence by students on authority. That’s bad. Second, we can’t add more time to the meetings, so doing more examples means either going through them in less detail or else using examples that are overly simple. In the first case, we have less time for questions and deep thought, and therefore more passivity and dependence. In the latter, we have examples that lack the conceptual clarity and generalizability to be of any use when solving a real problem. That’s also bad.

I think what students really want when they ask for “more” examples is to be more involved with the examples they get. They want more than a false sense of competency that often follows a well-constructed example that’s performed by an expert — they want first-hand experience with constructing solutions. That’s great! But this can be done poorly too, for instance through group work that throws students into the deep end of a difficult problem and leaves them to fend for themselves. What’s needed is a middle ground between pure lecture and pure group work.

For me, that middle ground is peer instruction. I’ve been using PI with all my classes this semester, but especially in Calculus 2 it seems to provide a happy medium for students and for me.

Here’s an example PI question from a unit on sequences:

Students were asked to think quietly by themselves for one minute, then vote using clickers. This slide followed a 10-minute minilecture that included a graphical interpretation of what “convergence” and “divergence” of sequences means, the technical definition of convergence, and five small examples of sequences and their convergence and divergence behaviors. When I say “small” I mean all five could be done in less than three minutes. So this wasn’t a lot of practice.

The correct response here is (d), Diverges, because the sequence does not get arbitrarily close to a single numerical value as \(n \to \infty\). On the first vote, the votes were 8% for response (a), 20% for response (c), and 72% for response (d). That 28% off the right answer was a little bothersome, so I just showed the histogram of clicker responses and had students get into groups of 2 or 3, with the instructions that they were to convince the other members of their group that they are correct. (Or if they are all in agreement, come up with a solid verbal explanation.) On the second vote, 100% of the class voted (d). All we needed was a quick debrief of why that answer was right, and we’re done. (That was for one section of Calculus 2. My other section had very similar numbers on that question.)

This PI question was better than an example worked at the board because the students were directly involved with it and because it highlights the concept at hand without getting bogged down in calculations. In subsequent problems, we could refer back to this clicker question and get a lot of use out of it. For example, students often mistake being divergent for being unbounded, but whenever that came up, we’d just say, “Remember that clicker question where the sequence was just \(-1, 1, -1, 1, \dots\)?” Yes, it does take longer than an example at the board — a PI question will take a minimum of 3 minutes for thinking, voting, and peer instruction without factoring in any debriefing. But the investment in time pays off in terms of having a longer shelf-life. Students tend to remember PI questions that generate a lot of discussion, as opposed to lecture items in which they have no vested interest.

Here’s another PI question:

This is a “loaded question” item because there’s no definition of “best” in this context. On the first vote, about 59% of the students voted (c) for the Integral Test, and about 38% voted for either the Comparison test or the Limit Comparison Test. Interestingly, this was at the end of a class session on the two comparison tests, but students didn’t default to those two methods for their answers just because that’s what we were covering. I put students into groups to discuss/convince, and at the second vote, the numbers were 82% for the Integral Test and 18% for the Limit Comparison Test. At the end, I asked the Integral Test voters why they thought that was best — and they told me how to set up the integral and calculate it. Then I asked the Limit Comparison Test voters the same thing, and they told me how to set up and calculate the limit. I added the Comparison Test approach at the end because I happen to think that’s the easiest way to do this. So out of this one PI question actually came three examples.

I’m telling students that peer instruction is better than examples because, although we don’t do as many of them as we might do lectured examples, we go into them more deeply and they get their hands on them first.

What about you? Are you using peer instruction in your classes, or maybe some other way to leverage better examples into your classes with active learning? Let’s hear it in the comments.

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I am a mathematician and educator with interests in cryptology, computer science, and STEM education. I am affiliated with the Mathematics Department at Grand Valley State University in Allendale, Michigan. The views here are my own and are not necessarily shared by GVSU.

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